Evaluate the expression without using a calculator. $$ \log _{8} \frac{1}{8} $$ $$ \log _{8} \frac{1}{8}=\square $$

Question

UpStudy AI · Accepted Answer

1. Write the fraction as an exponent:  
   $$
   \frac{1}{8} = 8^{-1}
   $$

2. Substitute into the logarithm expression:  
   $$
   \log_{8}\left(8^{-1}\right)
   $$

3. Use the logarithm property:  
   $$
   \log_{a}(a^x) = x
   $$
   Here, $$a = 8$$ and $$x = -1$$.

4. Thus, we have:  
   $$
   \log_{8}\left(8^{-1}\right) = -1
   $$

So,  
$$
\log_{8} \frac{1}{8} = -1
$$
$$ \log_{8} \frac{1}{8} = -1 $$

Evaluate the expression without using a calculator.